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MATH1151 week10c1

# MATH1151 week10c1 - Week 10 Monday 12-1 Class Time allowed...

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Time allowed: 20 minutes. 1. (3 marks) It is known that the vector 2 0 1 k is in span 1 2 3 4 , 3 6 9 12 , 3 3 3 0 , 1 1 4 8 . (a) Determine the value of k . (b) Hence or otherwise, for the value of k determined in part (a) above, find values of λ 1 , λ 2 , λ 3 , λ 4 such that: 6 0 3 3 k = λ 1 1 2 3 4 + λ 2 3 6 9 12 + λ 3 3 3 3 0 + λ 4 1 1 4 8 . 2. (3 marks) We would like to fit the quadratic curve y = β 1 + β 2 x + β 3 x 2 to a set of points ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) by the method of least squares. (a) Write down the normal equations in matrix form. (b) For this dataset, n = 12 , x i = 22 , x 2 i = 11 , x 3 i = 6 , x 4 i = 1 , x 5 i = 0 . 5 , y i = 45 , x i y i = 38 , x 2 i y i = 22 , x i y 2 i = 35 , x 2 i y 2 i = 58. Show that β 1 = 1 , β 2 = 2 and β 3 = 1. 3. (4 marks) Suppose that ˜ v 1 = 1 0 0 1 , ˜ v 2 = 1 0 1 0 , ˜ v 3 = 1 1 1 1 and π = span( ˜ v 1 , ˜ v 2 ). (a) Find an orthogonal basis for π . (b) Hence, or otherwise, find proj π ˜ v 3 .

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MATH1151 week10c1 - Week 10 Monday 12-1 Class Time allowed...

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